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172 VIII DIFFERENTIAL CALCULUS

and therefore a* is differentiabk in R, and D(a^x) = cp(a) - a^x, where (p(a) ^ 0
if a / 1. Suppose a ^ 1, and let b be any number > 0; we can write

b^x =
and therefore, by (8.4.1)

in other words

(p(b) = (p(a) log_a b.

There is therefore one and only one number e > 0 such that (p(e) = 1,
namely e = a^1/(f>(a}; as DO*) = e^x > 0, e^x is strictly increasing (by (8.5.3)),
and hence e = e¹ > e° = 1. The function e^x is also written exp(^c) or exp x.
The function log_e x is written log x and it follows from (8.2.3) and (4.2.2)
that D(log x) = l/x for ;c> 0. Furthermore D(a^x) = log a • a^x.

PROBLEM

Study the variation of the functions

HP H" HH^: ('+f

for x > 0,p being a fixed arbitrary positive number; find their limits when x tends to + oo.

9. PARTIAL DERIVATIVES

Let/be a differentiate mapping of an open subset A of a Banach space
E into a Banach space F; D/is then a mapping of A into «£?(£; F). We say
that/is continuously differentiable in A if D/is continuous in A.

Suppose now E = E! x E₂. For each point (a_l9 a₂) e A we can consider
the partial mappings x_l ->f(x_l9 a₂) and x₂~*f(a_i,x₂) of open subsets of
E! and E₂ respectively into F. We say that at (a_l9 a₂), f is differentiable
with respect to the first (resp. second) variable if the partial mapping
^xi ->/(*i> ^a2) (resp. Jt₂->/(fli, JC₂)) is differentiable at a^ (resp. a₂); the
derivative of that mapping, which is an element of J5f(E_x; F) (resp. J$f(E₂; F))
is called the partial derivative of/at (a_v, a₂) with respect to the first (resp.
second) variable, and written &if(a_l9 a₂) (resp. D₂f(a_l9 a₂)).



	172 VIII DIFFERENTIAL CALCULUS and therefore a* is differentiabk in R, and D(a^x) = cp(a) - a^x, where (p(a) ^ 0 if a / 1. Suppose a ^ 1, and let b be any number > 0; we can write b^x = and therefore, by (8.4.1)

	in other words (p(b) = (p(a) log_a b. There is therefore one and only one number e > 0 such that (p(e) = 1, namely e = a^1/(f>(a}; as DO) = e^x* > 0, e^x is strictly increasing (by (8.5.3)), and hence e = e¹ > e° = 1. The function e^x is also written exp(^c) or exp x. The function log_e x is written log x and it follows from (8.2.3) and (4.2.2) that D(log x) = l/x for ;c> 0. Furthermore D(a^x) = log a • a^x.

	PROBLEM Study the variation of the functions HP H" HH^: ('+f for x > 0,p being a fixed arbitrary positive number; find their limits when x tends to + oo.

	9. PARTIAL DERIVATIVES Let/be a differentiate mapping of an open subset A of a Banach space E into a Banach space F; D/is then a mapping of A into «£?(£; F). We say that/is continuously differentiable in A if D/is continuous in A. Suppose now E = E! x E₂. For each point (a_l9 a₂) e A we can consider the partial mappings x_l ->f(x_l9 a₂) and x₂~f(a_i,x₂)* of open subsets of E! and E₂ respectively into F. We say that at (a_l9 a₂), f is differentiable with respect to the first (resp. second) variable if the partial mapping ^xi ->/(i> ^a2)* (resp. Jt₂->/(fli, JC₂)) is differentiable at a^ (resp. a₂); the derivative of that mapping, which is an element of J5f(E_x; F) (resp. J$f(E₂; F)) is called the partial derivative of/at (a_v, a₂) with respect to the first (resp. second) variable, and written &if(a_l9 a₂) (resp. D₂f(a_l9 a₂)).